Solids of Revolution: The Method of Shells

Objective: This demo builds a toolbox of teaching aids to illustrate various aspects of volume calculations using the method of shells. Several props are used to demonstrate the geometric ideas of "shells" and the notion of "nesting" of shells to obtain an approximation of the volumes of solids of revolution. A collection of animations is included which can be run on a number of platforms.
Level: This demo can be presented in any course in which calculation of volumes of solids of revolution using the method of shells is introduced.
Prerequisites: Students should be familiar with areas of planar regions using approximations by Riemann sums and limits that lead to the definite integral. Prior knowledge of basic ideas concerning volumes is useful.
Platform: Interactive MATLAB routines and Mathematica notebooks that illustrate volumes modeled using the method of shells are given. A gallery of animations that run in web browsers is given. Several physical props are suggested that are useful for helping students understand the geometric concepts involving the method of shells.
Instructor's Notes: For this demo, we focus on regions bounded by the graph of a continuous function y = f(x) on the interval [a,b], the vertical lines x = a and x = b. For the illustrations we also require that f(x) is nonnegative over [a,b]. Several regions of this type are shown in Figure 1.
Figure 1. 
In addition to regions of the type illustrated in Figure 1, we consider regions that are bounded between the graphs of two functions y = f(x) and y = g(x) such as the shaded region in Figure 2.
Figure 2. 
The region is revolved about the yaxis to generate a solid of revolution. The object of the problem is to first approximate, then compute (using limits of Riemann sums) the volume of the solid of revolution.
Although the volume of any solid of revolution can be computed using a slicing approach (such as disks or washers), it can require a dualmethod approach. In Figure 3a, an animation shows the approximating slices required when revolving the region bounded by y = x^{2}+1, y = 0, x = 0 and x = 1. In the lower part of the solid, the approximating slice is a disk; in the part above, the slice is a washer. Thus, the integral representation for the volume requires two integrals.
In some cases, a slicing approach is not very practical (or would be often be done incorrectly by students). An example illustrating this difficulty is the solid generated by revolving the graph of y = sin(x) on the interval about the yaxis. Although one can visualize the rectangles that generate the washers (Figure 3b) and observe that integration would be with respect to y, the inside radius of the k^{th} washer is
,
but the outside radius is
,
a fact that would elude many students. The volume of the resulting solid of revolution can be more easily found using the method of shells.
Figure 3a. 
Figure 3b. 
The method of shells is fundamentally different from the method of disks. The method of disks involves slicing the solid perpendicular to the axis of revolution to obtain the approximating elements. However, the method of shells fills the solid with cylindrical shells in which the axis of the cylinder is parallel to the axis of revolution. An animation illustrating the construction of such a cylindrical shell for the example in Figure 3b is shown in Figure 4. Note that the approximation process involves generating a partition; the animation shows the generation of a partition and construction of one of the approximating shells.
Figure 4. 
Some Useful Props
The method of shells is based upon filling the solid with cylindrical shells. To motivate the ideas central to the method of shells, there are several props that can be useful as visualization tools.
Central to the development of the method of shells is the idea of nesting or layering of the approximating elements. The notion of nesting can be introduced using the layers of an onion (Figure 5).
Figure 5. 
The onion is made up of these layers so the volume of the onion could be computed if we could add the volumes of each of the layers.
Another useful prop to illustrate this idea is a set of Matroyska dolls. In Figure 6, we see that the hollow dolls of varying sizes nest together compactly.
Figure 6. 
Once students understand how the approximating elements will fit together, we need to show what kind of approximating element we will use and how to compute its volume.
The snack crackers, Combos (Figure 7), are excellent props to illustrate the cylindrical shells. Sharing these crackers with students is a fun way to help them develop the visualization skills that will help give meaning to the approximation scheme.
Figure 7. 
To illustrate the computation of the volume of a cylindrical shell, a paper towel roll or toilet paper roll is handy (Figure 8).
Figure 8. 
A nice intuitive approach to approximating the volume can be obtained by cutting the roll open and flattening it. We see that the shell can be quite thin, and in that case, the inner and outer radii are close to being equal. The volume is approximately the area of the face multiplied by the thickness, as illustrated in Figure 9.
Figure 9. 
To be more precise, we take an alternate approach to compute the exact volume of the cylindrical shell as shown in Figure 10. The inside radius is in red; the outside radius in blue.
Figure 10. 
We're now ready to put these ideas into the context of volumes of solids of revolution. We can think of a shell as being "generated" by a rectangle with height f(x) and width Dx. If the graph of y = f(x) is revolved about the yaxis, the radius of a shell is measured from the axis of revolution, given by x. The height of a shell is f(x), and the thickness is Dx, shown in Figure 11.
Figure 11. 
In general, the volume of the k^{th} shell is
.
By allowing the thickness of each shell to become very small and summing up the volumes of the shells, we obtain the definite integral representation for the method of shells:
.
The animation in Figure 12 illustrates the steps involved with the shell method for computing the volume of the solid of revolution generated by revolving the region in the first quadrant bounded by the graph of y = sin(x) and the xaxis about the yaxis. First, the region is partitioned and a typical shell is drawn. Approximating halfshells are drawn. To complete the visualization, the approximating shells are produced. After the approximating shells are drawn, the solid of revolution is generated.
Figure 12. 
When the solid is formed by revolving the region between the graphs of
y = f(x) and y = g(x), where f(x) > g(x), about the yaxis, the height of the rectangle is given by h = f(x)g(x) (Figure 13).
y = f(x) and y = g(x), where f(x) > g(x), about the yaxis, the height of the rectangle is given by h = f(x)g(x) (Figure 13).
Figure 13. 
Thus the integral representation is
.
The animation in Figure 14 illustrates the steps involved with the shell method for computing volume of the solid of revolution generated by revolving the region in the first quadrant between and about the yaxis.
Figure 14. 
A gallery of sample demos for illustrating the shell method for volumes of solids of revolution is available by clicking on SHELLMETHODGALLERY. These animations can be used by instructors in a classroom setting or by students to aid in acquiring a visualization background relating to the steps of the shell method. The demos provide a variety of animations for some common examples. Also included is a stepbystep narrative script of the displays in the animations.
Classroom Activities:
 An informative discussion about using props as teaching aids is in Carol Critchlow's paper 'A prop is worth ten thousand words', Mathematics Teacher, Vol. 92, No. 1, January 1999.
 For a description of a good handson project involving volumes of revolution see Judith Schimmel's paper 'A New Spin on Volumes of Solids of Revolution', Mathematics Teacher, Vol. 90, No.9, December 1997. This work incorporates modeling and employs both a calculator and computer software.
Technology Resources:
There are a variety of resources that employ calculators or software for illustrating and computing volumes of solids of revolution. Following is a sample of such resources which can be located using a search engine. We have chosen ones that relate to the method of shells.
 A program for the TI89 is available at http://www.ticalc.org/archives/files/fileinfo/73/7338.html
 A lab using Maple to visualize and calculate volumes of solids of revolution is available at http://www.mapleapps.com/powertools/CalcII/html/L2solidRevolution.html
 An example from the Visual Calculus collection is available at http://archives.math.utk.edu/visual.calculus/5/volumes.6/index.html.
 Mathematica notebooks, developed by Lila Roberts, that were used to generate the animations in this demo and the accompanying gallery can be downloaded. Versions that generate the shells for various regions are included along with suggestions on how the files can be modified for additional examples.
 A MATLAB utility, developed by David R. Hill, illustrates steps of the method of shells for several examples. To see an animation of how the utility works and to download the Mfile, .
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